Golden Spiral Collages

Over at Randomly Inspired!, one of my other projects, I have just released something that I think you’ll find interesting…and perhaps even beautiful. The golden ratio has been appearing in my life a lot lately: from the Fibonacci sequence to paintings by da Vinci to flowers and the shape of galaxies, this form is found in so many places in nature and art. I knew I wanted to put together something using this relationship, and it finally came together when I saw a golden spiral tattoo on someone at the gym.

I’m not into tattoos, but I will give a nod to this particular one—so much classier than an ex-girlfriend’s name or a poorly translated Chinese character. I had a flash of inspiration and thought golden spirals would make a great match for the amazing public domain photo collection at the Flickr Commons (oh, the things you think of while on the stair climber!).

So, I wrote some code that turns random photos from the Commons into golden spiral collages. You can see them here:

Golden Spiral Collages

I find that the proportions of the golden ratio are a pleasing way to view these repeated images. The form also creates interesting relationships between the larger picture and its components, as in this example:

Sometimes, the spiraling plays with your sense of proportion and depth, making it seem like the photo is a single view, continuing on and on in the repeated frames.

Programming and Troubleshooting Notes:

• I write and test code on my laptop before pushing it “live.” A great thing about Ruby on Rails is that my development environment is almost exactly the same as the Heroku hosting platform. Rails ensures that the version of Ruby and all the helper applications are the same in both places, so there’s a high probability that a problem encountered in development will also appear in production (and vice versa). I’ve written my fair share of bugs, but when I’ve finally nailed a piece of code in development, it usually works in production. However, this time around I encountered an interesting problem where the Image Magick gem would refuse to fetch remote content when it was deployed on the live site. It turns out that Heroku had disabled this feature (for security reasons), thereby creating a slight difference between the two environments. While finding a workaround was easy, the hard part was discovering this discrepancy!
• The final version of the program that generates the golden spirals was surprisingly brief. I guess this ties into why the self-repeating form is so prevalent in nature: it’s a very efficient way to encode a growth pattern. However, there was a lot of trial and error before I found the simplest way to turn the golden spiral concept into code. First, I tried the outside=>inside approach (maybe I should’ve begun with a middle-out scheme), starting with the largest block of the golden spiral and working towards smaller blocks. The problem with this method is that individual pixels can’t be cut in the middle, meaning that the precision of this method becomes sloppier (yes, very scientific term) as the size of the building blocks decrease. The solution was to reverse the process, starting with the smallest blocks and adding the larger ones to it. I got the idea by looking at this graph:

  

  This construction scheme was aided by the fact that Image Magick will scale an image to any resolution—even as small as 1×1! This meant that the photo collages could be built as Fibonacci spirals, an approximation of the golden ratio which gets more accurate as the blocks get larger. Here’s a table showing how successive pairs of the Fibonacci sequence converge on the golden ratio:

  Fibonacci Numbers		Ratio
  1	1	1
  1	2	2
  2	3	1.5
  3	5	1.66666666666667
  5	8	1.6
  8	13	1.625
  13	21	1.61538461538462
  21	34	1.61904761904762
  34	55	1.61764705882353
  55	89	1.61818181818182
  89	144	1.61797752808989
  144	233	1.61805555555556
  233	377	1.61802575107296
  377	610	1.61803713527851
  610	987	1.61803278688525
  987	1597	1.61803444782168
  1597	2584	1.61803381340013
  2584	4181	1.61803405572755
  4181	6765	1.61803396316671
  6765	10946	1.6180339985218
  10946	17711	1.61803398501736
  17711	28657	1.6180339901756
  28657	46368	1.61803398820533
  46368	75025	1.6180339889579
  75025	121393	1.61803398867044
  121393	196418	1.61803398878024
  196418	317811	1.6180339887383
  317811	514229	1.61803398875432
  514229	832040	1.6180339887482
  832040	1346269	1.61803398875054
  1346269	2178309	1.61803398874965

References

• Header Image: Golden spiral collage of  “Amaryllis” from the Provincial Archives of Alberta on Flickr Commons.